Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 19 x + 169 x^{2} - 779 x^{3} + 1681 x^{4}$ |
| Frobenius angles: | $\pm0.155792084339$, $\pm0.294748354530$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.641693.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 9 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1053$ | $2789397$ | $4780334637$ | $7996556848293$ | $13424781593403648$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $23$ | $1659$ | $69359$ | $2829875$ | $115874518$ | $4750136235$ | $194754255775$ | $7984926832483$ | $327381962051255$ | $13422659496787374$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=31x^6+3x^5+4x^4+11x^3+21x^2+24x$
- $y^2=23x^6+20x^5+31x^4+3x^3+10x+6$
- $y^2=19x^6+31x^5+7x^4+29x^3+28x+22$
- $y^2=12x^6+26x^5+13x^4+17x^3+28x^2+22x+2$
- $y^2=17x^6+28x^5+23x^4+9x^3+22x^2+28x+29$
- $y^2=24x^6+28x^5+3x^4+10x^3+22x^2+14x+20$
- $y^2=12x^6+30x^5+30x^4+23x^3+12x^2+17x+11$
- $y^2=14x^6+12x^5+2x^4+21x^3+30x^2+38x+3$
- $y^2=35x^6+40x^5+2x^4+30x^3+32x^2+29x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is 4.0.641693.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.41.t_gn | $2$ | (not in LMFDB) |